Question

A radioactive element decays to one third of its initial amount in time '$t$'. What fraction of the element would be left after $0.5t$ time?

• A. $\frac{1}{\sqrt{3}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\sqrt{\frac{2}{3}}$

Answer 1

The correct option is A $\frac{1}{\sqrt{3}}$

Using integrated rate law.

$\Rightarrow \lambda t=ln\frac{{\left[N\right]}_o}{{\left[N\right]}_t}$

Using the given information,

$\lambda t=ln\left(\frac{N_o}{N/3}\right)\Rightarrow \lambda t=ln3$

$\Rightarrow$Multiplying both sides by $\frac{1}{2}$.

$\lambda \left(\frac{t}{2}\right)=\frac{1}{2}ln3$

$\to \frac{\lambda t}{2}=ln\sqrt{3}$

Comparing with rate law.

$\lambda \left(t/2\right)=ln\left(\frac{N_o}{N_o/\sqrt{3}}\right)$

$\therefore \frac{1}{\sqrt{3}}$times the initial amount will be present after $0.5t$.